**6. 4f-EDD in rare-earth hexa-borides**

4f-EDD analysis has become more and more important since many rare-earth compounds with interesting physical properties, such as high-temperature super conductors, have been found. The EDD of rare-earth hexa-borides were investigated since they are famous for their properties related to the Kondo effect.

#### **6.1. XAO analysis of CeB6**

Ce and B atoms are at the body-centre and on the edges of the cubic unit cell with the B6 regular octahedra at the corners as shown in Fig. 7. Ce is in an *Oh* crystal field and B atoms are on 4-fold axes. CeB6 is a typical dense Kondo material with Kondo-temperature *Tk*=2 K. The magnetic moment decreases with that of temperature and vanishes below TK. However, Kondo effect starts above room temperature (Ōnuki et al. 1984). Thus, change of EDD below room temperature is interesting although our diffractometer did not permit to measure below 2K. Intensities were measured at 100, 165, 230 and 298 K aiming to separate the two kinds of asphericity of EDD stated in 5.1 and 5.2 (Tanaka & Ōnuki, 2002). MD was avoided by scan with the program *IUANGL* (Tanaka et al., 1994). Crystal was shaped into a sphere with a radius 36 μm to reduce absorption, extinction and MD.

**Figure 7.** Crystal structure of CeB6

Spin-orbit interaction was introduced into the XAO analysis. Four-fold degenerate 4f5/2<sup>8</sup> orbitals were taken from Table 4 (b) of *I* as,

286 Recent Advances in Crystallography

4 2.0 0.71(4) 0.04(32

**6. 4f-EDD in rare-earth hexa-borides** 

properties related to the Kondo effect.

**6.1. XAO analysis of CeB6** 

**Figure 7.** Crystal structure of CeB6

by  When G-C formalism was applied to an iron complex, it removed the peaks equally well as the multipole refinement did (Mallinson, et al., 1988). Our AHV analysis is based on the classical thermodynamics and does not have such a problem, though the p.d.f. function *exp[-V(***u***)/kBT]* is not integrable (Scheringer, 1977) at a place far from the nucleus. However, we can apply it safely as long as it is applied within the region where the sum of the third and fourth terms in (18) is much smaller than *kBT*. The condition is usually fulfilled in the investigations of EDD.

2 2 2 1 1.0 0.97(11) 0.21(59) 0.23(216) 0.57(59) 0.10(79)

*ddddd*

**0.63(22) 0.61(34)**

**0.84(43) 0.22(15)**

) 0.10(29) 0.34(41) 0.06(10)

 

**0.97(6)**

*<sup>i</sup> <sup>i</sup> xy z yz zx xy i n*

3 2.0 0.81(5) 0.15(52) 0.44(44) 0.06(22)

4f-EDD analysis has become more and more important since many rare-earth compounds with interesting physical properties, such as high-temperature super conductors, have been found. The EDD of rare-earth hexa-borides were investigated since they are famous for their

Ce and B atoms are at the body-centre and on the edges of the cubic unit cell with the B6 regular octahedra at the corners as shown in Fig. 7. Ce is in an *Oh* crystal field and B atoms are on 4-fold axes. CeB6 is a typical dense Kondo material with Kondo-temperature *Tk*=2 K. The magnetic moment decreases with that of temperature and vanishes below TK. However, Kondo effect starts above room temperature (Ōnuki et al. 1984). Thus, change of EDD below room temperature is interesting although our diffractometer did not permit to measure below 2K. Intensities were measured at 100, 165, 230 and 298 K aiming to separate the two kinds of asphericity of EDD stated in 5.1 and 5.2 (Tanaka & Ōnuki, 2002). MD was avoided

scan with the program *IUANGL* (Tanaka et al., 1994). Crystal was shaped into a sphere

**0.93(13)**

2 2.0 1.37(7) 0.08(28) 0.12(17) 0.19(19) 0.10(26)

5 2.0 0.90(5) 0.15(76) 0.22(45) 0.42(85)

**Table 1.** 3d-orbital parameters cik's of Cu2+. Significant ones in thick lines

with a radius 36 μm to reduce absorption, extinction and MD.

**0.75(34)**

$$\Psi\_1^{4f} = \sqrt{\frac{5}{6}}\boldsymbol{\nu}\_1 + \sqrt{\frac{1}{6}}\boldsymbol{\nu}\_{5'}\,\Psi\_2^{4f} = \sqrt{\frac{1}{6}}\boldsymbol{\nu}\_2 + \sqrt{\frac{5}{6}}\boldsymbol{\nu}\_{6'}\,\Psi\_3^{4f} = \boldsymbol{\nu}\_{3'}\,\Psi\_4^{4f} = \boldsymbol{\nu}\_{4'} \tag{25}$$

where *<sup>i</sup> i*1 to 6) are basis functions in Table 1(c) of *I*. In the similar way *5p1/2, 5p3/2* and *4f5/27* and *4f7/26,7,<sup>8</sup>* orbitals were introduced into the XAO analysis.

**Figure 8.** Difference density on (100) plane around Ce at (a) 100 K, (b) 165 K, (c) 230 K, (d) 298 K. Contours are the same as those in Fig. 2.


**Table 2.** Orbital and AHV parameters. n(X), and *κ*(X) are a number of electrons and *κ* of the orbital X. n(x10*-2*), κ *(*x-2*,* q*iijj* (x10*-19* J*Å-4*). *5p3/2,1/2* and *2s* are fully occupied. *n(2px)=n(2py)* and *κ(2px)*=*κ*(*2py*). Unit of Ce-B is Å. \*:n(*Γ7*) is 4.2(3.6)x10-2 only at 298 K.

#### *6.1.1. XAO analysis of CeB6 below room temperature*

After spherical-atom refinement for ions, Ce3+ and B0.5-, the populations of the orbitals were refined keeping the sum of them equal to that of nuclear charges. When populations of them exceeded one/two or became negative, they were fixed to one/two or zero. In our program *QNTAO* (Tanaka, 2000) each sub-shell is treated as an pseudo-atom sharing atomic parameters with the other sub-shells. It enables us to analyze the EDD of non-stoichiometric atoms, since valence electrons can be treated independently from the core electrons. Basis functions in Table 1 of *I* are automatically assigned to each AO labelled with *2l+1* and the number of basis functions. Since high symmetry fixes *cik'*s in (1), the other orbital parameters and atomic parameters, including AHV were refined. Electron density around Ce on (001) after spherical-atom refinement is shown in Fig. 8. The peak heights around Ce increase from *0.6, 1.2 to 2.0 e*Հ*-3*on lowering the temperature to 165 K and reduce to *0.6* eՀ-3 at *100 K*. After XAO analysis they were removed, exhibiting they were due to the *4f*-electrons. Parameters of AO's and AHV are listed in Table 2. No electrons were found in *4f7/2* orbitals but B-*2s* and Ce-*5p* are always fully occupied. *n(4f5/2Γ7)* has value 0.042(36) only at 298 K but *4f5/2Γ8* is occupied at the four temperatures. It agrees with the previous experiments (Zirngiebl et al., 1984;Sato et al., 1984). Shorter Ce-B at 298 K than that at 230 K seems to be correlated to *4f5/2Γ7*. It extends along 111 or to the centre of B6 octahedron making them extend to reduce electrostatic repulsion. Expanded *2px* still has electrons. Total number of *4f*-elctrons mainly composed of *4f5/2Γ8* are 0.98(11), 0.77(8), 0.61(7) and 0.52(6) at 298, 230, 165 and 100 K. They decrease with temperature. Table 2 tells *2px,y* and *4f* electrons are transferred to *2pz*, main contributor to the shortest covalent B-B bonds between B6 octahedra (called as B-Bout).

Electron accumulation at *B-Bout* enhances *q1111* and *q1122* of Ce at 165 K. They change signs and become more larger at 100 K. Negative and positive *q1111* at 165 K and 100K indicates that vibration of Ce at body-centre to the face-centre is favorable at 165 K, and becomes unfavorable at 100 K. This is the reason for the enhanced peak in Fig. 8(b) and reduced one in (a). Negative *q1122* at 100 K indicates the attractive force to the edge centre increased by the accumulated *2pz* electrons. Since sum of the mean radii of B-*2p* and Ce-*5p* in free space, 2.2048+1.7947=3.9995 Å (Mann, 1968) is longer than Ce-B distance in Table 2, a slight expansion/contraction of them affects the potential seriously. Since the 5p orbitals are fully occupied, EDD's of them are spherical. B-*2px* is closer than *2pz* to the spherically distributed Ce-*5p3/2* electrons. (*5p3/2*) exhibits slight expansion on lowering the temperature. Expansion and contraction of *5p3/2*and *4f5/2<sup>7</sup>* in Table 2 reduces the potential of *4f* orbitals lying closer to the nucleus. However, it contradicts to the decrease of *4f* population. It indicates that the system resists against losing *4f* electrons from Ce by making the potential of Ce more stable. It reminds us of the Le Chatelier's law: the system resists against the change. Then why are *4f* electrons allowed to flow out of Ce in spite of the reduced potential energy of the *<sup>8</sup>* state? The enhanced AHV at 165 and 100 K produced new ways of vibration along 100 and 110 directions increasing the ways to attain the energy of the system. Therefore, the enhanced AHV at lower temperatures means an increase in entropy. Since the electron transfer itself from Ce to (B-B)out increases entropy, it cannot be stopped.

The XAO analysis of CeB6 was applied to the weak-field model. The crystal structure, EDD, electron populations, expansion/contraction parameters and AHV parameters were quite consistent with each other at different temperatures. Therefore, the 4f-EDD is concluded to be measured and analyzed successfully.

#### *6.1.2. 4f population inversion and fully occupied 5d states at 430 K*

288 Recent Advances in Crystallography

*0.6, 1.2 to 2.0 e*

Ce-*5p3/2* electrons.

be measured and analyzed successfully.

and contraction of *5p3/2*and *4f5/2*

Հ

*QNTAO* (Tanaka, 2000) each sub-shell is treated as an pseudo-atom sharing atomic parameters with the other sub-shells. It enables us to analyze the EDD of non-stoichiometric atoms, since valence electrons can be treated independently from the core electrons. Basis functions in Table 1 of *I* are automatically assigned to each AO labelled with *2l+1* and the number of basis functions. Since high symmetry fixes *cik'*s in (1), the other orbital parameters and atomic parameters, including AHV were refined. Electron density around Ce on (001) after spherical-atom refinement is shown in Fig. 8. The peak heights around Ce increase from

XAO analysis they were removed, exhibiting they were due to the *4f*-electrons. Parameters of AO's and AHV are listed in Table 2. No electrons were found in *4f7/2* orbitals but B-*2s* and Ce-*5p* are always fully occupied. *n(4f5/2Γ7)* has value 0.042(36) only at 298 K but *4f5/2Γ8* is occupied at the four temperatures. It agrees with the previous experiments (Zirngiebl et al., 1984;Sato et al., 1984). Shorter Ce-B at 298 K than that at 230 K seems to be correlated to *4f5/2Γ7*. It extends along 111 or to the centre of B6 octahedron making them extend to reduce electrostatic repulsion. Expanded *2px* still has electrons. Total number of *4f*-elctrons mainly composed of *4f5/2Γ8* are 0.98(11), 0.77(8), 0.61(7) and 0.52(6) at 298, 230, 165 and 100 K. They decrease with temperature. Table 2 tells *2px,y* and *4f* electrons are transferred to *2pz*, main contributor to the

Electron accumulation at *B-Bout* enhances *q1111* and *q1122* of Ce at 165 K. They change signs and become more larger at 100 K. Negative and positive *q1111* at 165 K and 100K indicates that vibration of Ce at body-centre to the face-centre is favorable at 165 K, and becomes unfavorable at 100 K. This is the reason for the enhanced peak in Fig. 8(b) and reduced one in (a). Negative *q1122* at 100 K indicates the attractive force to the edge centre increased by the accumulated *2pz* electrons. Since sum of the mean radii of B-*2p* and Ce-*5p* in free space, 2.2048+1.7947=3.9995 Å (Mann, 1968) is longer than Ce-B distance in Table 2, a slight expansion/contraction of them affects the potential seriously. Since the 5p orbitals are fully occupied, EDD's of them are spherical. B-*2px* is closer than *2pz* to the spherically distributed

the nucleus. However, it contradicts to the decrease of *4f* population. It indicates that the system resists against losing *4f* electrons from Ce by making the potential of Ce more stable. It reminds us of the Le Chatelier's law: the system resists against the change. Then why are *4f*

enhanced AHV at 165 and 100 K produced new ways of vibration along 100 and 110 directions increasing the ways to attain the energy of the system. Therefore, the enhanced AHV at lower temperatures means an increase in entropy. Since the electron transfer itself

The XAO analysis of CeB6 was applied to the weak-field model. The crystal structure, EDD, electron populations, expansion/contraction parameters and AHV parameters were quite consistent with each other at different temperatures. Therefore, the 4f-EDD is concluded to

electrons allowed to flow out of Ce in spite of the reduced potential energy of the

shortest covalent B-B bonds between B6 octahedra (called as B-Bout).

from Ce to (B-B)out increases entropy, it cannot be stopped.

*-3*on lowering the temperature to 165 K and reduce to *0.6* eՀ-3 at *100 K*. After

(*5p3/2*) exhibits slight expansion on lowering the temperature. Expansion

*<sup>7</sup>* in Table 2 reduces the potential of *4f* orbitals lying closer to

*<sup>8</sup>* state? The

The electron population at 298 K exhibited slightly occupied *4f5/2<sup>7</sup>*. Since the energy gap between *4f5/28* and *<sup>7</sup>* was reported to be 530-560 K (Loewenhaupt et al., 1985; Zirngiebl et al., 1984), XAO analysis at 430 K was performed to observe more electrons in *<sup>7</sup>* (Makita et al., 2007). For details of the high-temperature diffraction equipment, see PhD thesis of Makita (Makita, 2008). Scattering factors of Ce were evaluated from relativistic AO's calculated by the program *GRASP* (Dyall et al., 1989). Difference density around Ce on (001) after the spherical-atom refinement is shown in Fig. 9(a). The 4f-peaks closest to Ce elongate along <110> in contrast to those below room temperature in Fig. 8. They are surrounded by the four peaks of 0.65 eÅ-3, which extend along <100>, and disappeared after the analysis of the 4f peaks. However, the population of *4f5/2Γ8* and *<sup>7</sup>* were 0.06(3) and 0.36(11). Higher temperature reverses the populations of *4f5/28* and *<sup>7</sup>*. The peaks outside the *4f*-peaks, called *5d* peaks, remained almost unchanged. Accordingly *5d5/27* and *<sup>8</sup>* orbitals were further refined. The population of *5d5/2Γ8, n(5d5/2Γ*, exceeded 1.0 and was fixed. *n(4f5/28, 7)* and *n(5d5/28)* were 0.06(2), 0.37(1) and 1.0, while *5d5/2<sup>7</sup>* was vacant. R factor was reduced from 1.25 to 1.16 %. The peaks outside the *4f* peaks reduced slightly from 0.65 to 0.59 eÅ-3 in Fig. 10(b). Since *5d* orbitals extend in a large area, in contrast to 4f orbitals, slight change of EDD is significant. Thus electron densities around *5d*–peaks in Figs. 9 (a) and (b) were numerically integrated to give 3.41 and 1.57 electrons. XAO analysis explained 54 % of the *5d-*peaks. Therefore *d5/2<sup>8</sup>* is concluded to be occupied, though peaks still remained. It may be ascribed to the inaccurate *5d-AO*'s (Claiser et al., 2004).

**Figure 9.** Differnece density around Ce at 430 K on (100) after (a) spherical-atom refinement and (b) final residual density. Contours at 0.2 eÅ-3. Negative and zero contours in dotted and thick lines.

Why are *5d* orbitals occupied? The energy level of the Ce-*5d* and B-*2p* calculated by Liberman et al., (1971) and Mann (1967) are -0.63 and -0.62 a.u. They are very close compared to the Ce-4f level at -0.75 a.u. Thus B-2p electrons are transferred to 5d levels predominantly according to the first-order perturbation theory. The radial distribution functions of the relevant AO's are illustrated in Fig. 10. Putting the origin of B at *r*=3.045 Å, *2p* was drawn to the reverse side. *5d* and *2p* overlap well, and it is expected that *2p* electrons which distribute all over the crystal through the network of B-B covalent bonds can be transferred most probably to *5d* orbitals. Electrons traveling in the crystal are expected to stay at *5d5/2Γ*8orbitals when they come close to Ce.

**Figure 10.** Radial distribution functions of Ca and B.

**Figure 11.** *4f5/2Γ<sup>8</sup>* (red) and *5d5/2Γ<sup>8</sup>* obritals.

**Figure 12.** 5d- and 4f-energy levels expected from the electron populations.

Why does *4f5/28-7* inversion occur? *5d5/2* orbitals are located outside of *4f5/2*8 having exactly the same symmetry (Fig. 11). Since *5d5/28* is fully occupied, the potential of the *4f5/28* orbital is enhanced. It is the reason for the population inversion of *4f5/2* and *<sup>7</sup>*. The energy level diagram expected from the electron populations obtained by the XAO analysis is shown in Fig.12. Since the quantization axes are defined parallel to <100>, T2g states locate higher than Eg. *5d3/2Γ* orbital closely related to *T2g* is located highest among the 5d orbitals as shown in Fig. 12. From the populations obtained, 2p electrons seem to be transferred to *5d5/2*8 orbitals, directly or first to *5d3/2<sup>8</sup>* and then to *5d5/28*.

#### *6.1.3. Electron configuration at 340 and 535 K*

290 Recent Advances in Crystallography

**Figure 10.** Radial distribution functions of Ca and B.

**Figure 11.** *4f5/2Γ<sup>8</sup>* (red) and *5d5/2Γ<sup>8</sup>* obritals.

**Figure 12.** 5d- and 4f-energy levels expected from the electron populations.

In order to confirm the *5d*-occupation in CeB6 at 430 K, XAO analyses were done at 340 and 535 K. The difference densities, as well as electron configurations, are different from each other as shown in Fig. 13. Since the electrons are continually transferred between *2p* and *5d* orbitals, the crystal field seems to change with temperature resulting in different directions of the 4f and 5d peaks in Fig. 13. However, it is confirmed that *5d* orbitals in CeB6 are occupied above room temperature (Makita et al., 2008).

**Figure 13.** Bird-eye view of difference densities around Ce at (a) 340, (b) 430 and (c) 535 K. Contours as in Fig. 9 except the interval of 0.1 eÅ-3. Frames are parallel to <100>axes.

#### **6.2. XAO analysis of SmB6**

SmB6 formally has five 4f electrons. In order to extend XAO analysis to multi 4f-electron system, EDD of SmB6 was measured at 100, 165, 230 and 298 K (Funahashi, et al., 2010). It is interesting to see how physical properties as a Kondo insulator are explained by the XAO analysis. The shadow in the difference density at 230 K in Fig. 14 specifies roughly the area of *5d* peaks. *5p3/2, 4f5/28,<sup>7</sup>* and AHV were refined reducing the *4f* peaks in Fig. 14(a). However, since *4f*-peaks still remained along <100>, the populations of *4f7/26,7,<sup>8</sup>* were refined but only *6* orbital which extends along <100> more sharply than *4f5/2<sup>8</sup>* survived. Since peaks (peak A) remained in the *5d*-area, *5d5/28,7* and *5d3/2<sup>8</sup>* were added in the refinement and only *5d5/2<sup>8</sup>* orbital, which stems from *eg* orbital of the storing field model had population. Fig. 14(b) exhibits that peak A is reduced from 0.43 to 0.17 eÅ-3 and 4f-peaks are almost deleted. Final parameters are listed in Table 3.

The electron configuration in Table 3 is correlated to the physical properties of SmB6 as follows: (a) SmB6 is a Kondo insulator. Its electric resistivity increases gradually like semiconductors below room temperature and begins to increase steeply below 30 K with a decrease in temperature. It also begins to increase like metals above room temperature (Ueda & Onuki, 1998). *2p* and *5d3/2Γ8* orbitals consist of the conduction band (Kimura et al., 1990). Since B-*2pz* extends along the edge, it does not overlap with *5d3/2Γ<sup>8</sup>* effectively and does not seem to contribute to the band. The population of *2px* in Table 3 as well as the resistivity steadily increases on lowering temperature. n(*5d5/2Γ8)* also increases from 230 K. The increase of populations indicates that of localized electrons. It may be correlated to the increase of the resistivity. (b) *4f7/2Γ6* are vacant only at 100 K, although they are occupied at the other three temperatures. It may be correlated to the band gap between the *4f* states, which is reported to start developing between 150 and 100 K (Souma, et al., 2002). (c) Among the *5d* orbitals only the *5d5/28* are occupied. Since *5d5/2<sup>8</sup>* orbitals correspond to eg in the strong field model as illustrated in Fig. 1, it agrees with the band calculation of LaB6 by Harima (1988), reporting that *5d-eg* and *2p* of B consist of the conduction band.

**Figure 14.** Difference density at 230 K after (a)spherical-atom refinement and (b) XAO analysis. Contours as in Fig. 13.


**Table 3.** Temperature dependence of electron populations of Sm *4f*, *5d* and B *2p* orbitals.

## **7. Bright future for X-ray crystallography**

EDD investigation was limited up to 3*d*-transition-metal complexes. However, XAO analysis made EDD investigations of rare-earth compounds as well as non-stoichiometric ones possible. Its application to organic compounds can be attained when it is developed to X-ray molecular orbital analysis (XMO). Since the least-squares method stated in 2.2 was formulated for MO models, XMO analysis will be accomplished in a near future. When actinoid compounds become our targets, however, MD-effect is expected to be so much that the avoidance of it by rotation is impossible and the correction for it will be inevitable.

CeB6 is a possible quantum-material to emit UV light when electrons in *5d5/2Γ8* could be transferred to *4f5/2Γ8*, as the investigation at 430 K revealed. The d-f transition is a permitted one by quantum mechanics. Since the 5d-occupation is found in the ground state of CeB6, some external force is necessary to make the transition occur. A electron populations in CeB6 and SmB6 found by the XAO analysis demostrate the importance of the EDD analysis based on quantum-mechanical orbitals.

As discussed in 5.3, the aspherical properties of EDD and AHV are separated better by the method based on classical Boltzmann statistics than the G-C method. However, recent development of neutron diffraction will make it possible to get intrinsic ADP's and use them as known parameters in X-ray EDD analyses. It will improve XAO analysis of rare-earth complexes and makes the XMO analysis surer and easier.

The accuracy of X-ray structure-factor measurements has been improved so much that every crystallographer will investigate EDD easily as a part of their X-ray structure analysis. The top-up operation with constant incident beam intensity at SR facilities has improved the accuracy of the structure factor measurements from 1 % to 0.1 %. Future of X-ray diffraction is bright.
